Some curvature estimates for Riemannian manifolds equipped with foliations of rank 2

7 Pages·0.12 MB·English

by  Pawel Walczak

SOME CURVATURE ESTIMATES
FOR RIEMANNIAN MANIFOLDS
EQUIPPED WITH FOLIATIONS OF RANK 2
6
9 Pawel Walczak
9
1
December 30, 1995

n
a
J
Abstract. Some curvature estimates are derivedfrom geometrical data concerning

4
quasi-conformality properties of some commuting linearly independent vector fields

on a compact Riemannian manifold.
1
v
1
0
0
1 1. Introduction. Calculating the rank, i.e. the maximal number of pointwise
0 linearly independent commuting vector fields, of a manifold M is a well known
6
problem in differential topology. If rankM = k, then M admits a locally free
9
/ action of Rk. Such actions (and foliations generated by them) are classified up to
a
g the either topological or differentiable type in several cases ([AC], [AS], [CRW],
-
g [RR], etc.). Also, for some pairs (k,n), n-dimensional closed manifolds of rank k
d are classified. For example, if k = n 1, then M is a Tm-bundle over a torus Tn−m
: −
v for some m ([CR], [R], [RRW], etc.).
i
X On the other hand, in several geometric situations (like infinitesimal isometries
r [Be] or infinitesimal conformal transformations on closed Riemannian manifolds of
a
positive sectional curvature, see [W] and [Y], for instance) vector fields have to
vanish at some points or have to be somewhere collinear whenever commute. The
result presented here is of this sort: Assuming the existence of two commuting
linearly independent vector fields X and Y on a Riemannian manifold (M,g) we
estimate the curvature of M in terms of some geometric data (like the divergence,
the norm of the Ahlfors operator) obtained from the R2-action generated by the
flows of X and Y. More precisely, our result reads as follows.
Theorem. For any ǫ > 0 there exists a constant K(ǫ) > 0 such that any closed
n-dimensional (n 2) Riemannian manifold of sectional curvature K > K(ǫ)
M
admits no (C1,ǫ)-q≥uasi-conformal R2-actions.
Roughly speaking, a vector field X is C1-quasi-conformal if the Ahlfors operator
S on (M,g) evaluated at X has the C1-norm small enough. An action φ of R2
is (C1,ǫ)-quasi-conformal if all the vector fields Z (u R2) given by Z (p) =
u u
∈
(t φ(tu,p))˙(0) (p M) are (C1,ǫ)-quasi-conformal. For the precise definition of
7→ ∈
infinitesimal (C1,ǫ)-quasi-conformal transformations, see Section 3.
1991 Mathematics Subject Classification. 53 C 20.
Supported by the KBN grant no 2P30103604
Typeset by AMS-TEX
The proof of the Theorem is given in Sections 2 - 5 below. In general, it follows
the line of that in [Y]. Several identities in there which hold for infinitesimal con-
formal transformations are replaced by inequalities produced from the geometric
data related to the vector fields under consideration. At some points, the argument
differs essentially. For instance, instead of studying the eigenvalues and eigenvec-
tors of some linear operator A we estimate the inner products of A(v) with some
particular vector y. Section 6 contains an application of the estimates derived be-
fore: We obtain an inequality which involves the Gaussian curvature of the orbits
of a locally free (C1,ǫ)-quasi-conformal action of R2 and their second fundmental
form.
Remark. Our Theorem together with the mentioned above results on the rank
of closed manifolds justify to some extend the following Yang’s remark [Y]: One
might expect that any two commuting vector fields on a closed manifold of positive
sectional curvature must be linearly dependent at some points, in other words,
if M is closed and has positive sectional curvature, then rankM 1. However,
≤
the problem of calculating (or, estimating) the rank of all manifolds which admit
positively curved Riemannian metrics is probably pretty difficult. First, there is a
lot of examples of positively curved closed manifolds of different topological types
([AW], [E1], [E2], [KS], etc.) and still new examples arrive (see [Ba] and [T] for
instance). Second, even if M is simply connected, the curvature of M is δ-pinched
withδ close enough to 1 (forexample, δ = 0.681)and, therefore, M is diffeomorphic
to the standard sphere Sn [S], the rank of M is unknown in odd dimensions. Of
course, rankSn = 0 when n is even. Moreover, rankS3 = 1 [L] and there are no
locally free actions of S1 R on any standard sphere [ASc].
×
2. A preliminary lemma. Let X be an arbitrary vector field on a Riemannian
manifold (M,g), p M and X(p ) = 0.
0 0
∈ 6
Lemma. For any vector v T M, v = 0, orthogonal to X(p ) there exists a
∈ p0 6 0
vector field V in a neighbourhood U of p such that
0
(1) V(p ) = v, V,X = 0, [V,X] = θ X and V = a X(p )
0 v 0
h i · ∇ ·
for some θ C∞(U) and a R.
∈ ∈
Proof. In a neighbourhood U of p0, there exists a chart φ = (x,y,z1,...,zn−2)
(n = dimM) such that X = ∂ , v = ∂ (p ) and ∂ = 0. Find a function
∂x ∂y 0 ∇v∂y
f C∞(U) such that V = ∂ +f ∂ is orthogonal to X. V satisfies (1) with
∈ ∂y · ∂x
∂f ∂f
(2) θ = and a = (p ). (cid:3)
0
−∂x ∂y
3. Infinitesimal (C1,ǫ)-quasi-conformal transformations. Given ǫ 0, a
≥
vector field X on a Riemannian manifold (M,g) will be called an infinitesimal
(C1,ǫ)-quasi-conformal transformation (an infinitesimal (C1,ǫ)-QCT, for short) if
2
(3) SX(p) ǫ X(p) and SX(p) ǫ X(p)
k k ≤ k k k∇ k ≤ k k
at any point p of M.
2
Here S is the Ahlfors operator on M. Recall that SX is a traceless symmetric
2-form on M defined by
(4) SX = g f g,
X X
L − ·
2
where denotes the Lie derivation with respect to X, f = divX and n =
LX X n
dimM. Equivalently, SX can be defined by
(5) SX(V,W) = X,W + V, X f V,W ,
V W X
h∇ i h ∇ i− ·h i
where V and W are arbitary vector fields and is the Levi-Civita connection on
∇
(M,g). Recall also that SX describes the rate of quasi-conformality of the flow
k k
maps(φ )ofX,namely,anymapφ isk -quasi-conformalwithk exp( t SX ),
t t t t
≤ | |·k k
being the supremum norm here [P].
k·k
Remark. Observe that our notion of quasi-conformality differs from the standard
one. Usually, a vector field X is said to be k-quasi-conformal just when SX k.
k k ≤
Also, observe that if the C1-norm of X is small,
∇
2 2
(6) X δ X and X δ X ,
k∇ k ≤ ·k k k∇ k ≤ ·k k
then X is (C1,ǫ)-quasi-conformal with ǫ = 4δ. Finally, note that if X is an infini-
tesimal (C1,ǫ)-QCT then, for any c = 0, cX is an infinitesimal (C1,ǫ)-QCT as well,
and if X is an infinitesimal (C1,ǫ)-Q6 CT on (M,g) and g′ = c2g for some c R ,
+
then X is an infinitesimal (C1,ǫ′)-QCT on (M,g′) with ǫ′ = c−1ǫ. ∈
4. First estimates. AssumenowthatX andY arepointwiselinearlyindependent
commutingvectorfieldsonacompactRiemannianmanifoldM suchthatu X+u Y
1 2
is an infinitesimal (C1,ǫ)-QCT for any u and u R.
1 2
For any p M and t S1 = R/2πZ put ∈
∈ ∈
(7) Z(p,t) = cost X(p)+sint Y(p).
· ·
The function h : M S1 R given by h(p,t) = Z(t)(p) 2 is continuous, therefore
× → k k
it attains its minimum at a point (p ,t ) of M S1. Since all the fields Z( ,t) are
0 0
× ·
(C1,ǫ)-quasi-conformal, we may assume without losing generality that that t = 0.
0
Let x = X(p ), y = Y(p ) and v T M be an unit vector. The standard
0 0 ∈ p0
analysis of partial derivatives of h at p shows that
0
(8) v X,X = 0, x,y = 0,
h i h i
2
(9) V X,X (p ) 0,
0
h i ≥
(10) y x
k k ≥ k k
and
V2 X,X (p ) v X,Y
0
(11) d = h i h i 0,
(cid:12) v X,Y y 2 x 2(cid:12) ≥
(cid:12) h i k k −k k (cid:12)
(cid:12) 3 (cid:12)
(cid:12) (cid:12)
where V is an arbitrary vector field on a neighbourhood of p , V(p ) = v.
0 0
Now, take any unit vector v x and extend it to a vector field V satisfying the
⊥
conditions of Lemma in Section 2. Then, at the point p as above,
0
2
(12) SX(x,x) = f (p ) x , therefore f (p ) ǫ x ,
X 0 X 0
− ·k k | | ≤ k k
2
(13) SX(x,v) = x X,V X,[X,V] = θ(p ) x , therefore θ(p ) ǫ
0 0
− h i−h i k k | | ≤
and
2
(14) SX(v,v) = 2 x, V f (p ) = 2a x f (p ), therefore a x ǫ.
v X 0 X 0
− h ∇ i− − k k − k k ≤
Moreover, X,x = 0and X,v = x, V = x, X x,[X,V](p )
x x x v 0
h∇ i h∇ i −h ∇ i −h ∇ i−h i
= θ(p ) x 2. From (13) it follows that
0
·k k
2
(15) X ǫ x .
x
k∇ k ≤ k k
Also, since SX(v,v) = x V,V f (p ), we get from (3) and (12) that
X 0
h i−
(16) X V,V (p ) 2ǫ x .
0
| h i | ≤ k k
Passing to the covariant derivative SX we observe that
∇
2
( SX)(v,v) = X V,V (p ) Xf (p ) f (p )X V,V (p )
x 0 X 0 X 0 0
(17) ∇ h i − − h i
2SX( V,v),
x
− ∇
2 2 2
(18) ( SX)(x,v) = vθ x SX( V,v) θ(p ) x a SX(x,x),
v x 0
∇ ·k k − ∇ − k k − ·
and
2 2
(19) ( SX)(x,x) = X X,X (p ) Xf (p ) x 2SX( X,x).
x 0 X 0 x
∇ h i − ·k k − ∇
Combining identities (17) - (19) and using inequalitites (3) and (12) - (16) we arrive
at
2 2 −2 2 2 2
(20) X V,V (p ) 2vθ x x X X,X (p ) 12ǫ x .
0 0
h i − ·k k −k k h i ≤ k k
(cid:12) (cid:12)
(cid:12) (cid:12)
All the inequalities above will be applied in the next Sections to get some cur-
vature estimates.
5. Estimates involving sectional curvature. Let us keep all the notation
introduced in Section 4 and assume that the sectional curvature of M is positive.
(Otherwise, the statement of the Theorem holds trivially.) Also, denote by A the
linear transformation of T M given by
p0
(21) Aw = X
w
∇
and choose a vector v in such a way that
(22) Av = by for some b R.
∈
4
⊥ ⊥ ⊥
(Such a vector exists. Since A : x x , either Ker A x is nontrivial and
{⊥ } → { } |{ }
Av = 0 for some v x or A x is an isomorphism.)
⊥ |{ }
As before, extend v to a vector field V defined in a neighbourhood of p as in our
0
Lemma. From the definition of the curvature tensor R on (M,g) we obtain easily
the identity
2 2 2 2
V X,X (p )+X V,V (p ) = 2K (x v) x +2 Av
0 0 M
(23) h i h i − ∧ k k k |k
2
+2vθ x 2θ(p ) Ax,v ,
0
·k k − h i
where
−2
(24) K (x v) = x R(v,x)x,v
M
∧ k k h i
is the sectional curvature of M in the direction of the plane spanned by x and v.
Since both X and Y are infinitesimal (C1,ǫ)-QCTs,
(25) Y,v + x, Y ǫ x y
x v
|h∇ i h ∇ i| ≤ k k·k k
and
(26) X,v + y, X 2ǫ x y .
y v
|h∇ i h ∇ i| ≤ k k·k k
Since X and Y commute, Y = X and (25) together with (26) imply that
x y
∇ ∇
2
(27) v X,Y 2b y 3ǫ x y .
| h i− k k | ≤ k k·k k
Consequently,
2 2 4 3 2 2 2
(28) (v X,Y ) 4b y +12ǫb x y +9ǫ x y .
− h i ≤ − k k k k·k k k k k k
The determinant d in (11) satisfies
2 2 2 2 2
d =( y x )( 2K(x z) x X V,V (p )+2 Av
0
(29) | | −k k − ∧ k k − h i k k
2 2
2θ(p ) Ax,v +2vθ x ) (v X,Y ) .
0
− h i ·k k − h i
Since the sectional curvature of M is positive, combining inequalities (9), (13), (15),
(20) and (28), and using identity (22) we obtain the inequality
2 2 2 2 2 2 2 4
0 d ( y x )(14ǫ x +2b y ) b y
(30) ≤ ≤ k k −k k k k k k − k k
3 2 2 2
+12ǫb x y +9ǫ x y .
k k·k k k k k k
Substituting s = b x −1 y , dividing both sides of (30) by y 2 and performing
k k k k k k
some other elementary transformations one can see easily that (30) implies
2 2
(31) 2s +12ǫs+23ǫ 0.
− ≥
Therefore,
1
(32) s (3+ √82)ǫ x
≤ 2 k k
and
1
(33) b (3+ √82)ǫ x y −1.
≤ 2 k k·k k
Applying formula (22) once again we can see that
1 1
−2 2 2 2
K(x v) = x ( V X,X (p ) X V,V (p )+ Av
0 0
∧ k k −2 h i − 2 h i k k
(34) −2 2 2 2 2
+vθ θ(p ) Ax,v ) x 6ǫ x +b y
0
− h i ≤ k k k k k k
ǫ2(36.5+3√82) 63.66615(cid:0)541...ǫ2 (cid:1)
≤ ≈
Therefore, the statement of the Theorem holds with
(35) K(ǫ) = ǫ2(36.5+3√82). (cid:3)
5
Corollary [Y]. Any two commuting conformal vector fields on a closed Riemann-
ian manifold of positive sectional curvature must be linearly dependent at least at
one point.
6. Estimates involving Gaussian curvature of the leaves. Let be a folia-
F
tion of a compact positively curved manifold M defined by a locally free action of
R2. Then dimF = 2 and the tangent bundle of is spanned by two commuting
F
linearly independent vector fields X and Y. These fields as well as all the linear
combinationsu X+u Y, u R, becomeinfinitesimal(C1,ǫ)-QCTsforsomeǫ 0.
1 2 i
Therefore, with this ǫ, is d∈efined by a (C1,ǫ)-quasi-conformal action of R2. ≥
F
Consider fields X and Y, and find p M as before. Let v be a unit vector
0
∈
tangent to and orthogonal to x = X(p ), say v = y/ y . Then
0
F k k
(36) Av = by +B(x,v),
where B is the second fundamental form of . The argument analogous to that of
F
Section 5 shows that
1
(37) b x y −1 (3+ √82)ǫ+ B(p ) ,
0
≤ k k·k k (cid:18) 2 k k(cid:19)
where x = x/ x . Applying formula (22) as in Section 5 we can get the estimate
1
k k
3
(38) K (x v) ǫ2(36.5+3√82)+ǫ√2(3+0.5√82) B(p ) + B(p ) 2.
M 0 0
∧ ≤ k k 2k k
Finally, the classical Gauss equation reads
2
(39) KF(p0) = KM(x v)+ B(x1,x1),B(v,v) B(x1,v) ,
∧ h i−k k
where KF is the Gaussian curvature of the leaves of and x1 = x/ x . Therefore,
F k k
we can conclude by the following.
Proposition. The Gaussian curvature KF of a foliation defined by a locally
free (C1,ǫ)-quasi-conformal action of R2 on a closed positiveFly curved Riemannian
manifold M satisfies at some point p of M the inequality
0
(40) KF(p0) ǫ2(36.5+3√82)+(6+√82) B(p0) + B(p0) 2
≤ k k k k
with B being the second fundamental form of . (cid:3)
F
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Instytut Matematyki, Uniwersytet L o´dzki, Banacha 22, 90238 L o´d´z, Poland
E-mail address: pawelwal@ krysia.uni.lodz.pl
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